Definition

The term Hausdorff measures is used for a class of outer measures (introduced for the first time by Hausdorff in [Ha]) on subsets of a generic metric space $(X,d)$, or for their restrictions to the corresponding measurable sets.

Let $(X,d)$ be a metric space. In what follows, for any subset $E\subset X$, ${\rm diam}\, (E)$ will denote the diameter of $E$.

The $\mathcal{H}^\alpha_\delta$ defined above are outer measures and they are called Hausdorff premeasures by some authors. Moreover, in \eqref{e:hausdorff_m} the infimum can be taken over open coverings or closed coverings without changing the result.

Generalizations

The definition of the Hausdorff measures is just a special case of a more general construction due to Caratheodory, which starting from a generic (nonnegative) set function $\nu$ with $\nu (\emptyset) =0$ builds an outer measure $\mu$ (we refer to Outer measure for a decription of Caratheodory's method). A generalization of the usual Hausdorff measures replaces $\omega_\alpha ({\rm diam}\, (E_i))^\alpha$ in \ref{e:hausdorff_m} with $h ({\rm diam}\, (E_i))$, where $h: \mathbb R^+\to \mathbb R^+$ is a nondecreasing function (often called gauge function). See for instance [Ma].

The construction of Caratheodory allows for several other outer measures in the Euclidean space, most of which coincide with the Hausdorff $k$-dimensional measures for $C^1$ submanifolds when $k$ is an integer, but differ on general sets. One example is the Favard measure, also called integralgeometric measure. See [Fe] and [KP].

In some common generalizations of the Hausdorff measures one restricts the class of admissible coverings in \ref{e:hausdorff_m}. For instance one can use coverings by balls (and the resulting outer measure is then called spherical Hausdorff measure) or by cylinders (cylindrical Hausdorff measure).

Measure-theoretic properties

The Hausdorff measures $\mathcal{H}^\alpha$ satisfy Caratheodory's criterion. Therefore, the $\sigma$-algebra of $\mathcal{H}^\alpha$-measurable sets (see Outer measure for the definition) contains the Borel sets (i.e. $\mathcal{H}^\alpha$ is a Borel outer measure). The Hausdorff measures are
also Borel regular, in the sense that, for any set $A\subset X$ there is a Borel set $B\supset A$ with $\mathcal{H}^\alpha (B) = \mathcal{H}^\alpha (A)$ (see Corollary 4.5 in [Ma]).

Remark 5
The premeasures $\mathcal{H}^\alpha_\delta$ do not satisfy Caratheodory's criterion and, moreover, they are not necessarily Borel outer measures: this property fails already in the Euclidean spaces (see [Si]).

$\mathcal{H}^n$ for $n$ integer

For $n$ integer the Hausdorff measures are suitable measure-theoretic generalizations of the concept of $n$-dimensional volume of a smooth Riemannian manifold.

The counting measure

In any metric space $(X,d)$ and for any set $E\subset X$, $\mathcal{H}^0 (E)$ equals the cardinality of $E$ if $E$ is a finite set and it equals infinity if not. $\mathcal{H}^0$ is called, therefore, the counting measure.

Length

In any metric space $(X,d)$, if $\gamma: [0,1]\to X$ is an injective Lipschitz function, then $\mathcal{H}^1 (\gamma ([0,1])$ is the length of the curve (see Rectifiable curve for the relevant definition).

$n$-dimensional volume

On the euclidean space $\mathbb R^n$ $\mathcal{H}^n$ coincides with the Lebesgue outer measure (see Theorem 2 in Section 2.2 of [EG]). More generally, in a Riemannian manifold $M$ of dimension $n$, $\mathcal{H}^n$ coincides with the standard volume. Thus, If $\Sigma$ is a $C^1$ submanifold of $\mathbb R^N$ of dimension $n$, then $\mathcal{H}^n (\Gamma)$ is the usual $n$-dimensional volume of $\Gamma$. In this case a useful tool to compute the Hausdorff measure is the Area formula.

Rectifiable sets

For several applications, the class of Borel sets of $\mathbb R^N$ with finite $\mathcal{H}^n$ measure is too large to be an appropriate generalization of smooth $n$-dimensional surfaces. An intermediate class which has wide applications is that of rectifiable sets.

Relations to density

Especially in the euclidean space there is a strong link between various concepts of densities of measures and sets and the Hausdorff measures (see Density of a set). This relation, pioneered by Besicovitch and his school (cf. [Ro]), plays a fundamental role in Geometric measure theory (see for instance [Fe], [KP] or [Si]).

Relevance

Hausdorff measures play an important role in several areas of mathematics